Loogle!

Result

Found 630 declarations mentioning ContinuousAt. Of these, only the first 200 are shown.

ContinuousAt 📋 Mathlib.Topology.Defs.Filter
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) (x : X) : Prop
ContinuousAt.mapClusterPt 📋 Mathlib.Topology.ClusterPt
{X : Type u} [TopologicalSpace X] {α : Type u_1} {F : Filter α} {u : α → X} [TopologicalSpace α] {a : α} (hf : ContinuousAt u a) (hu : ClusterPt a F) : MapClusterPt (u a) F u
MapClusterPt.continuousAt_comp 📋 Mathlib.Topology.ClusterPt
{X : Type u} [TopologicalSpace X] {Y : Type v} {α : Type u_1} {F : Filter α} {u : α → X} {x : X} [TopologicalSpace Y] {f : X → Y} (hf : ContinuousAt f x) (hu : MapClusterPt x F u) : MapClusterPt (f x) F (f ∘ u)
continuousAt_id 📋 Mathlib.Topology.Continuous
{X : Type u_1} [TopologicalSpace X] {x : X} : ContinuousAt id x
continuousAt_id' 📋 Mathlib.Topology.Continuous
{X : Type u_1} [TopologicalSpace X] (y : X) : ContinuousAt (fun x => x) y
continuousAt_const 📋 Mathlib.Topology.Continuous
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x : X} {y : Y} : ContinuousAt (fun x => y) x
Continuous.continuousAt 📋 Mathlib.Topology.Continuous
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {x : X} (h : Continuous f) : ContinuousAt f x
continuous_iff_continuousAt 📋 Mathlib.Topology.Continuous
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} : Continuous f ↔ ∀ (x : X), ContinuousAt f x
Filter.EventuallyEq.continuousAt 📋 Mathlib.Topology.Continuous
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {x : X} {y : Y} (h : f =ᶠ[nhds x] fun x => y) : ContinuousAt f x
ContinuousAt.iterate 📋 Mathlib.Topology.Continuous
{X : Type u_1} [TopologicalSpace X] {x : X} {f : X → X} (hf : ContinuousAt f x) (hx : f x = x) (n : ℕ) : ContinuousAt f^[n] x
ContinuousAt.tendsto 📋 Mathlib.Topology.Continuous
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {x : X} (h : ContinuousAt f x) : Filter.Tendsto f (nhds x) (nhds (f x))
ContinuousAt.congr 📋 Mathlib.Topology.Continuous
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {x : X} {g : X → Y} (hf : ContinuousAt f x) (h : f =ᶠ[nhds x] g) : ContinuousAt g x
continuousAt_congr 📋 Mathlib.Topology.Continuous
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {x : X} {g : X → Y} (h : f =ᶠ[nhds x] g) : ContinuousAt f x ↔ ContinuousAt g x
ClusterPt.map 📋 Mathlib.Topology.Continuous
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {x : X} {lx : Filter X} {ly : Filter Y} (H : ClusterPt x lx) (hfc : ContinuousAt f x) (hf : Filter.Tendsto f lx ly) : ClusterPt (f x) ly
ContinuousAt.comp' 📋 Mathlib.Topology.Continuous
{X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y} {g : Y → Z} {x : X} (hg : ContinuousAt g (f x)) (hf : ContinuousAt f x) : ContinuousAt (fun x => g (f x)) x
ContinuousAt.comp 📋 Mathlib.Topology.Continuous
{X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y} {x : X} {g : Y → Z} (hg : ContinuousAt g (f x)) (hf : ContinuousAt f x) : ContinuousAt (g ∘ f) x
mem_closure_image 📋 Mathlib.Topology.Continuous
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s : Set X} {x : X} (hf : ContinuousAt f x) (hx : x ∈ closure s) : f x ∈ closure (f '' s)
ContinuousAt.comp_of_eq 📋 Mathlib.Topology.Continuous
{X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y} {x : X} {y : Y} {g : Y → Z} (hg : ContinuousAt g y) (hf : ContinuousAt f x) (hy : f x = y) : ContinuousAt (g ∘ f) x
ContinuousAt.eventually_mem 📋 Mathlib.Topology.Continuous
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {x : X} (hf : ContinuousAt f x) {s : Set Y} (hs : s ∈ nhds (f x)) : ∀ᶠ (y : X) in nhds x, f y ∈ s
ContinuousAt.preimage_mem_nhds 📋 Mathlib.Topology.Continuous
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {x : X} {t : Set Y} (h : ContinuousAt f x) (ht : t ∈ nhds (f x)) : f ⁻¹' t ∈ nhds x
continuousAt_def 📋 Mathlib.Topology.Continuous
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {x : X} : ContinuousAt f x ↔ ∀ A ∈ nhds (f x), f ⁻¹' A ∈ nhds x
not_continuousAt_of_tendsto 📋 Mathlib.Topology.Continuous
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {l₁ : Filter X} {l₂ : Filter Y} {x : X} (hf : Filter.Tendsto f l₁ l₂) [l₁.NeBot] (hl₁ : l₁ ≤ nhds x) (hl₂ : Disjoint (nhds (f x)) l₂) : ¬ContinuousAt f x
IsOpenMap.map_nhds_eq 📋 Mathlib.Topology.Maps.Basic
{X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) {x : X} (hf' : ContinuousAt f x) : Filter.map f (nhds x) = nhds (f x)
Topology.IsInducing.continuousAt_iff 📋 Mathlib.Topology.Maps.Basic
{X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace Y] [TopologicalSpace X] [TopologicalSpace Z] (hg : Topology.IsInducing g) {x : X} : ContinuousAt f x ↔ ContinuousAt (g ∘ f) x
IsOpenMap.of_sections 📋 Mathlib.Topology.Maps.Basic
{X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (h : ∀ (x : X), ∃ g, ContinuousAt g (f x) ∧ g (f x) = x ∧ Function.RightInverse g f) : IsOpenMap f
Topology.IsOpenEmbedding.continuousAt_iff 📋 Mathlib.Topology.Maps.Basic
{X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (hf : Topology.IsOpenEmbedding f) {x : X} : ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x)
Topology.IsInducing.continuousAt_iff' 📋 Mathlib.Topology.Maps.Basic
{X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace Y] [TopologicalSpace X] [TopologicalSpace Z] (hf : Topology.IsInducing f) {x : X} (h : Set.range f ∈ nhds (f x)) : ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x)
IsOpenQuotientMap.continuousAt_comp_iff 📋 Mathlib.Topology.Maps.OpenQuotient
{X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y} (h : IsOpenQuotientMap f) {g : Y → Z} {x : X} : ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x)
Homeomorph.comp_continuousAt_iff 📋 Mathlib.Topology.Homeomorph.Defs
{X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (h : X ≃ₜ Y) (f : Z → X) (z : Z) : ContinuousAt (⇑h ∘ f) z ↔ ContinuousAt f z
Homeomorph.comp_continuousAt_iff' 📋 Mathlib.Topology.Homeomorph.Defs
{X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (h : X ≃ₜ Y) (f : Y → Z) (x : X) : ContinuousAt (f ∘ ⇑h) x ↔ ContinuousAt f (h x)
continuousAt_fst 📋 Mathlib.Topology.Constructions.SumProd
{X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {p : X × Y} : ContinuousAt Prod.fst p
continuousAt_snd 📋 Mathlib.Topology.Constructions.SumProd
{X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {p : X × Y} : ContinuousAt Prod.snd p
ContinuousAt.fst 📋 Mathlib.Topology.Constructions.SumProd
{X : Type u} {Y : Type v} {Z : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y × Z} {x : X} (hf : ContinuousAt f x) : ContinuousAt (fun x => (f x).1) x
ContinuousAt.snd 📋 Mathlib.Topology.Constructions.SumProd
{X : Type u} {Y : Type v} {Z : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y × Z} {x : X} (hf : ContinuousAt f x) : ContinuousAt (fun x => (f x).2) x
ContinuousAt.fst' 📋 Mathlib.Topology.Constructions.SumProd
{X : Type u} {Y : Type v} {Z : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Z} {x : X} {y : Y} (hf : ContinuousAt f x) : ContinuousAt (fun x => f x.1) (x, y)
ContinuousAt.fst'' 📋 Mathlib.Topology.Constructions.SumProd
{X : Type u} {Y : Type v} {Z : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Z} {x : X × Y} (hf : ContinuousAt f x.1) : ContinuousAt (fun x => f x.1) x
ContinuousAt.snd' 📋 Mathlib.Topology.Constructions.SumProd
{X : Type u} {Y : Type v} {Z : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : Y → Z} {x : X} {y : Y} (hf : ContinuousAt f y) : ContinuousAt (fun x => f x.2) (x, y)
ContinuousAt.snd'' 📋 Mathlib.Topology.Constructions.SumProd
{X : Type u} {Y : Type v} {Z : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : Y → Z} {x : X × Y} (hf : ContinuousAt f x.2) : ContinuousAt (fun x => f x.2) x
ContinuousAt.prodMk 📋 Mathlib.Topology.Constructions.SumProd
{X : Type u} {Y : Type v} {Z : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y} {g : X → Z} {x : X} (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (fun x => (f x, g x)) x
ContinuousAt.prodMap' 📋 Mathlib.Topology.Constructions.SumProd
{X : Type u} {Y : Type v} {W : Type u_1} {Z : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [TopologicalSpace W] {f : X → Z} {g : Y → W} {x : X} {y : Y} (hf : ContinuousAt f x) (hg : ContinuousAt g y) : ContinuousAt (Prod.map f g) (x, y)
continuousAt_prodMap_iff 📋 Mathlib.Topology.Constructions.SumProd
{X : Type u} {Y : Type v} {W : Type u_1} {Z : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [TopologicalSpace W] {f : X → Z} {g : Y → W} {x : X} {y : Y} : ContinuousAt (Prod.map f g) (x, y) ↔ ContinuousAt f x ∧ ContinuousAt g y
ContinuousAt.prodMap 📋 Mathlib.Topology.Constructions.SumProd
{X : Type u} {Y : Type v} {W : Type u_1} {Z : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [TopologicalSpace W] {f : X → Z} {g : Y → W} {p : X × Y} (hf : ContinuousAt f p.1) (hg : ContinuousAt g p.2) : ContinuousAt (Prod.map f g) p
ContinuousAt.comp₂ 📋 Mathlib.Topology.Constructions.SumProd
{X : Type u} {Y : Type v} {W : Type u_1} {Z : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [TopologicalSpace W] {f : Y × Z → W} {g : X → Y} {h : X → Z} {x : X} (hf : ContinuousAt f (g x, h x)) (hg : ContinuousAt g x) (hh : ContinuousAt h x) : ContinuousAt (fun x => f (g x, h x)) x
ContinuousAt.comp₂_of_eq 📋 Mathlib.Topology.Constructions.SumProd
{X : Type u} {Y : Type v} {W : Type u_1} {Z : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [TopologicalSpace W] {f : Y × Z → W} {g : X → Y} {h : X → Z} {x : X} {y : Y × Z} (hf : ContinuousAt f y) (hg : ContinuousAt g x) (hh : ContinuousAt h x) (e : (g x, h x) = y) : ContinuousAt (fun x => f (g x, h x)) x
continuousAt_subtype_val 📋 Mathlib.Topology.Constructions
{X : Type u} [TopologicalSpace X] {p : X → Prop} {x : Subtype p} : ContinuousAt Subtype.val x
continuousAt_apply 📋 Mathlib.Topology.Constructions
{ι : Type u_2} {A : ι → Type u_3} [T : (i : ι) → TopologicalSpace (A i)] (i : ι) (x : (i : ι) → A i) : ContinuousAt (fun p => p i) x
continuousAt_pi' 📋 Mathlib.Topology.Constructions
{X : Type u} {ι : Type u_2} {A : ι → Type u_3} [TopologicalSpace X] [T : (i : ι) → TopologicalSpace (A i)] {f : X → (i : ι) → A i} {x : X} (hf : ∀ (i : ι), ContinuousAt (fun y => f y i) x) : ContinuousAt f x
continuousAt_pi 📋 Mathlib.Topology.Constructions
{X : Type u} {ι : Type u_2} {A : ι → Type u_3} [TopologicalSpace X] [T : (i : ι) → TopologicalSpace (A i)] {f : X → (i : ι) → A i} {x : X} : ContinuousAt f x ↔ ∀ (i : ι), ContinuousAt (fun y => f y i) x
ContinuousAt.codRestrict 📋 Mathlib.Topology.Constructions
{X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {t : Set Y} (h1 : ∀ (x : X), f x ∈ t) {x : X} : ContinuousAt f x → ContinuousAt (Set.codRestrict f t h1) x
continuousAt_codRestrict_iff 📋 Mathlib.Topology.Constructions
{X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {t : Set Y} (h1 : ∀ (x : X), f x ∈ t) {x : X} : ContinuousAt (Set.codRestrict f t h1) x ↔ ContinuousAt f x
ContinuousAt.piMap 📋 Mathlib.Topology.Constructions
{ι : Type u_2} {A : ι → Type u_3} {B : ι → Type u_4} [T : (i : ι) → TopologicalSpace (A i)] [(i : ι) → TopologicalSpace (B i)] {f : (i : ι) → A i → B i} {x : (i : ι) → A i} (hf : ∀ (i : ι), ContinuousAt (f i) (x i)) : ContinuousAt (Pi.map f) x
ContinuousAt.update 📋 Mathlib.Topology.Constructions
{X : Type u} {ι : Type u_2} {A : ι → Type u_3} [TopologicalSpace X] [T : (i : ι) → TopologicalSpace (A i)] {f : X → (i : ι) → A i} [DecidableEq ι] {x : X} (hf : ContinuousAt f x) (i : ι) {g : X → A i} (hg : ContinuousAt g x) : ContinuousAt (fun a => Function.update (f a) i (g a)) x
ContinuousAt.matrixVecCons 📋 Mathlib.Topology.Constructions
{X : Type u} {Z : Type u_1} [TopologicalSpace X] [TopologicalSpace Z] {n : ℕ} {f : X → Z} {g : X → Fin n → Z} {x : X} (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (fun a => Matrix.vecCons (f a) (g a)) x
ContinuousAt.restrict 📋 Mathlib.Topology.Constructions
{X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s : Set X} {t : Set Y} (h1 : Set.MapsTo f s t) {x : ↑s} (h2 : ContinuousAt f ↑x) : ContinuousAt (Set.MapsTo.restrict f s t h1) x
ContinuousAt.restrictPreimage 📋 Mathlib.Topology.Constructions
{X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s : Set Y} {x : ↑(f ⁻¹' s)} (h : ContinuousAt f ↑x) : ContinuousAt (s.restrictPreimage f) x
ContinuousAt.finInit 📋 Mathlib.Topology.Constructions
{X : Type u} [TopologicalSpace X] {n : ℕ} {A : Fin (n + 1) → Type u_6} [(i : Fin (n + 1)) → TopologicalSpace (A i)] {f : X → (j : Fin (n + 1)) → A j} {x : X} (hf : ContinuousAt f x) : ContinuousAt (fun a => Fin.init (f a)) x
ContinuousAt.finTail 📋 Mathlib.Topology.Constructions
{X : Type u} [TopologicalSpace X] {n : ℕ} {A : Fin (n + 1) → Type u_6} [(i : Fin (n + 1)) → TopologicalSpace (A i)] {f : X → (j : Fin (n + 1)) → A j} {x : X} (hf : ContinuousAt f x) : ContinuousAt (fun a => Fin.tail (f a)) x
ContinuousAt.finSnoc 📋 Mathlib.Topology.Constructions
{X : Type u} [TopologicalSpace X] {n : ℕ} {A : Fin (n + 1) → Type u_6} [(i : Fin (n + 1)) → TopologicalSpace (A i)] {f : X → (j : Fin n) → A j.castSucc} {g : X → A (Fin.last n)} {x : X} (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (fun a => Fin.snoc (f a) (g a)) x
ContinuousAt.finInsertNth 📋 Mathlib.Topology.Constructions
{X : Type u} [TopologicalSpace X] {n : ℕ} {A : Fin (n + 1) → Type u_6} [(i : Fin (n + 1)) → TopologicalSpace (A i)] (i : Fin (n + 1)) {f : X → A i} {g : X → (j : Fin n) → A (i.succAbove j)} {x : X} (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (fun a => i.insertNth (f a) (g a)) x
ContinuousAt.finCons 📋 Mathlib.Topology.Constructions
{X : Type u} [TopologicalSpace X] {n : ℕ} {A : Fin (n + 1) → Type u_6} [(i : Fin (n + 1)) → TopologicalSpace (A i)] {f : X → A 0} {g : X → (j : Fin n) → A j.succ} {x : X} (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (fun a => Fin.cons (f a) (g a)) x
continuousAt_iff_punctured_nhds 📋 Mathlib.Topology.NhdsWithin
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {a : α} : ContinuousAt f a ↔ Filter.Tendsto f (nhdsWithin a {a}ᶜ) (nhds (f a))
continuousAt_of_not_accPt_top 📋 Mathlib.Topology.ContinuousOn
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {x : α} (h : ¬AccPt x ⊤) : ContinuousAt f x
continuousWithinAt_univ 📋 Mathlib.Topology.ContinuousOn
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α → β) (x : α) : ContinuousWithinAt f Set.univ x ↔ ContinuousAt f x
ContinuousAt.continuousWithinAt 📋 Mathlib.Topology.ContinuousOn
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} {x : α} (h : ContinuousAt f x) : ContinuousWithinAt f s x
continuousOn_of_forall_continuousAt 📋 Mathlib.Topology.ContinuousOn
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} (hcont : ∀ x ∈ s, ContinuousAt f x) : ContinuousOn f s
ContinuousAt.congr_of_eventuallyEq 📋 Mathlib.Topology.ContinuousOn
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f g : α → β} {x : α} (h : ContinuousAt f x) (hg : g =ᶠ[nhds x] f) : ContinuousAt g x
continuousAt_of_not_accPt 📋 Mathlib.Topology.ContinuousOn
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {x : α} (h : ¬AccPt x (Filter.principal {x}ᶜ)) : ContinuousAt f x
continuousWithinAt_compl_self 📋 Mathlib.Topology.ContinuousOn
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {x : α} : ContinuousWithinAt f {x}ᶜ x ↔ ContinuousAt f x
ContinuousOn.continuousAt 📋 Mathlib.Topology.ContinuousOn
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} {x : α} (h : ContinuousOn f s) (hx : s ∈ nhds x) : ContinuousAt f x
ContinuousWithinAt.continuousAt 📋 Mathlib.Topology.ContinuousOn
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} {x : α} (h : ContinuousWithinAt f s x) (hs : s ∈ nhds x) : ContinuousAt f x
IsOpen.continuousOn_iff 📋 Mathlib.Topology.ContinuousOn
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} (hs : IsOpen s) : ContinuousOn f s ↔ ∀ ⦃a : α⦄, a ∈ s → ContinuousAt f a
continuousWithinAt_iff_continuousAt 📋 Mathlib.Topology.ContinuousOn
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} {x : α} (h : s ∈ nhds x) : ContinuousWithinAt f s x ↔ ContinuousAt f x
ContinuousAt.comp_continuousWithinAt 📋 Mathlib.Topology.ContinuousOn
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {f : α → β} {s : Set α} {x : α} {g : β → γ} (hg : ContinuousAt g (f x)) (hf : ContinuousWithinAt f s x) : ContinuousWithinAt (g ∘ f) s x
ContinuousAt.comp_continuousWithinAt_of_eq 📋 Mathlib.Topology.ContinuousOn
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {f : α → β} {s : Set α} {x : α} {g : β → γ} {y : β} (hg : ContinuousAt g y) (hf : ContinuousWithinAt f s x) (hy : f x = y) : ContinuousWithinAt (g ∘ f) s x
continuousAt_prod_of_discrete_left 📋 Mathlib.Topology.ContinuousOn
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [DiscreteTopology α] {f : α × β → γ} {x : α × β} : ContinuousAt f x ↔ ContinuousAt (fun x_1 => f (x.1, x_1)) x.2
continuousAt_prod_of_discrete_right 📋 Mathlib.Topology.ContinuousOn
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [DiscreteTopology β] {f : α × β → γ} {x : α × β} : ContinuousAt f x ↔ ContinuousAt (fun x_1 => f (x_1, x.2)) x.1
ContinuousOn.union_continuousAt 📋 Mathlib.Topology.ContinuousOn
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {s t : Set α} {f : α → β} (s_op : IsOpen s) (hs : ContinuousOn f s) (ht : ∀ x ∈ t, ContinuousAt f x) : ContinuousOn f (s ∪ t)
Function.LeftInverse.map_nhds_eq 📋 Mathlib.Topology.ContinuousOn
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {g : β → α} {x : β} (h : Function.LeftInverse f g) (hf : ContinuousWithinAt f (Set.range g) (g x)) (hg : ContinuousAt g x) : Filter.map g (nhds x) = nhdsWithin (g x) (Set.range g)
continuousWithinAt_iff_continuousAt_domRestrict 📋 Mathlib.Topology.ContinuousOn
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α → β) {x : α} {s : Set α} (h : x ∈ s) : ContinuousWithinAt f s x ↔ ContinuousAt (s.domRestrict f) ⟨x, h⟩
continuousWithinAt_iff_continuousAt_restrict 📋 Mathlib.Topology.ContinuousOn
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α → β) {x : α} {s : Set α} (h : x ∈ s) : ContinuousWithinAt f s x ↔ ContinuousAt (s.domRestrict f) ⟨x, h⟩
ContinuousAt.comp₂_continuousWithinAt 📋 Mathlib.Topology.ContinuousOn
{α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] {f : β × γ → δ} {g : α → β} {h : α → γ} {x : α} {s : Set α} (hf : ContinuousAt f (g x, h x)) (hg : ContinuousWithinAt g s x) (hh : ContinuousWithinAt h s x) : ContinuousWithinAt (fun x => f (g x, h x)) s x
ContinuousAt.comp₂_continuousWithinAt_of_eq 📋 Mathlib.Topology.ContinuousOn
{α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] {f : β × γ → δ} {g : α → β} {h : α → γ} {x : α} {s : Set α} {y : β × γ} (hf : ContinuousAt f y) (hg : ContinuousWithinAt g s x) (hh : ContinuousWithinAt h s x) (e : (g x, h x) = y) : ContinuousWithinAt (fun x => f (g x, h x)) s x
continuousAt_iff_ultrafilter 📋 Mathlib.Topology.Ultrafilter
{X : Type u} {Y : Type v} {x : X} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} : ContinuousAt f x ↔ ∀ (g : Ultrafilter X), ↑g ≤ nhds x → Filter.Tendsto f (↑g) (nhds (f x))
Specializes.map_of_continuousAt 📋 Mathlib.Topology.Inseparable
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x y : X} {f : X → Y} (h : x ⤳ y) (hf : ContinuousAt f y) : f x ⤳ f y
Inseparable.map_of_continuousAt 📋 Mathlib.Topology.Inseparable
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x y : X} {f : X → Y} (h : Inseparable x y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) : Inseparable (f x) (f y)
SeparationQuotient.continuousAt_lift 📋 Mathlib.Topology.Inseparable
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x : X} {f : X → Y} {hf : ∀ (x y : X), Inseparable x y → f x = f y} : ContinuousAt (SeparationQuotient.lift f hf) (SeparationQuotient.mk x) ↔ ContinuousAt f x
SeparationQuotient.continuousAt_lift₂ 📋 Mathlib.Topology.Inseparable
{X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y → Z} {hf : ∀ (a : X) (b : Y) (c : X) (d : Y), Inseparable a c → Inseparable b d → f a b = f c d} {x : X} {y : Y} : ContinuousAt (Function.uncurry (SeparationQuotient.lift₂ f hf)) (SeparationQuotient.mk x, SeparationQuotient.mk y) ↔ ContinuousAt (Function.uncurry f) (x, y)
continuousAt_update_same 📋 Mathlib.Topology.Piecewise
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {x : α} [DecidableEq α] {y : β} : ContinuousAt (Function.update f x y) x ↔ Filter.Tendsto f (nhdsWithin x {x}ᶜ) (nhds y)
continuousAt_of_tendsto_nhds 📋 Mathlib.Topology.Separation.Basic
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T1Space Y] {f : X → Y} {x : X} {y : Y} (h : Filter.Tendsto f (nhds x) (nhds y)) : ContinuousAt f x
ContinuousAt.eventually_ne 📋 Mathlib.Topology.Separation.Basic
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T1Space Y] {g : X → Y} {x : X} {y : Y} (hg1 : ContinuousAt g x) (hg2 : g x ≠ y) : ∀ᶠ (z : X) in nhds x, g z ≠ y
continuousAt_update_of_ne 📋 Mathlib.Topology.Separation.Basic
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [T1Space X] [DecidableEq X] [TopologicalSpace Y] {f : X → Y} {x x' : X} {y : Y} (hne : x' ≠ x) : ContinuousAt (Function.update f x y) x' ↔ ContinuousAt f x'
ContinuousAt.ne_iff_eventually_ne 📋 Mathlib.Topology.Separation.Hausdorff
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y] {x : X} {f g : X → Y} (hf : ContinuousAt f x) (hg : ContinuousAt g x) : f x ≠ g x ↔ ∀ᶠ (x : X) in nhds x, f x ≠ g x
ContinuousAt.eventuallyEq_nhds_iff_eventuallyEq_nhdsNE 📋 Mathlib.Topology.Separation.Hausdorff
{X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y] {x : X} {f g : X → Y} (hf : ContinuousAt f x) (hg : ContinuousAt g x) [(nhdsWithin x {x}ᶜ).NeBot] : f =ᶠ[nhdsWithin x {x}ᶜ] g ↔ f =ᶠ[nhds x] g
Set.InjOn.exists_isOpen_superset 📋 Mathlib.Topology.Separation.Hausdorff
{X : Type u_3} {Y : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y] {f : X → Y} {s : Set X} (inj : Set.InjOn f s) (sc : IsCompact s) (fc : ∀ x ∈ s, ContinuousAt f x) (loc : ∀ x ∈ s, ∃ u ∈ nhds x, Set.InjOn f u) : ∃ t, IsOpen t ∧ s ⊆ t ∧ Set.InjOn f t
Set.InjOn.exists_mem_nhdsSet 📋 Mathlib.Topology.Separation.Hausdorff
{X : Type u_3} {Y : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y] {f : X → Y} {s : Set X} (inj : Set.InjOn f s) (sc : IsCompact s) (fc : ∀ x ∈ s, ContinuousAt f x) (loc : ∀ x ∈ s, ∃ u ∈ nhds x, Set.InjOn f u) : ∃ t ∈ nhdsSet s, Set.InjOn f t
continuousAt_iff_continuous_left'_right' 📋 Mathlib.Topology.Order.LeftRight
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [LinearOrder α] [TopologicalSpace β] {a : α} {f : α → β} : ContinuousAt f a ↔ ContinuousWithinAt f (Set.Iio a) a ∧ ContinuousWithinAt f (Set.Ioi a) a
continuousAt_iff_continuous_left_right 📋 Mathlib.Topology.Order.LeftRight
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [LinearOrder α] [TopologicalSpace β] {a : α} {f : α → β} : ContinuousAt f a ↔ ContinuousWithinAt f (Set.Iic a) a ∧ ContinuousWithinAt f (Set.Ici a) a
ContinuousAt.eventually_lt 📋 Mathlib.Topology.Order.OrderClosed
{α : Type u} {β : Type v} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α} [TopologicalSpace β] {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀) (hfg : f x₀ < g x₀) : ∀ᶠ (x : β) in nhds x₀, f x < g x
IsDenseInducing.extend_eq_at 📋 Mathlib.Topology.DenseEmbedding
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} [TopologicalSpace γ] [T2Space γ] (di : IsDenseInducing i) {f : α → γ} {a : α} (hf : ContinuousAt f a) : di.extend f (i a) = f a
IsDenseInducing.inseparable_extend 📋 Mathlib.Topology.DenseEmbedding
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} [TopologicalSpace γ] [R1Space γ] (di : IsDenseInducing i) {f : α → γ} {a : α} (hf : ContinuousAt f a) : Inseparable (di.extend f (i a)) (f a)
IsDenseInducing.tendsto_extend 📋 Mathlib.Topology.DenseEmbedding
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} [TopologicalSpace γ] (di : IsDenseInducing i) {f : α → γ} {a : α} (hf : ContinuousAt f a) : Filter.Tendsto f (nhds a) (nhds (di.extend f (i a)))
IsDenseInducing.continuousAt_extend 📋 Mathlib.Topology.DenseEmbedding
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} [TopologicalSpace γ] [T3Space γ] {b : β} {f : α → γ} (di : IsDenseInducing i) (hf : ∀ᶠ (x : β) in nhds b, ∃ c, Filter.Tendsto f (Filter.comap i (nhds x)) (nhds c)) : ContinuousAt (di.extend f) b
IsDenseInducing.extend_unique_at 📋 Mathlib.Topology.DenseEmbedding
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} [TopologicalSpace γ] [T2Space γ] {b : β} {f : α → γ} {g : β → γ} (di : IsDenseInducing i) (hf : ∀ᶠ (x : α) in Filter.comap i (nhds b), g (i x) = f x) (hg : ContinuousAt g b) : di.extend f b = g b
Dense.extend_eq_at 📋 Mathlib.Topology.DenseEmbedding
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {s : Set α} [T2Space β] (hs : Dense s) {f : ↑s → β} {x : ↑s} (hf : ContinuousAt f x) : hs.extend f ↑x = f x
Dense.continuousAt_extend 📋 Mathlib.Topology.DenseEmbedding
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {s : Set α} {f : ↑s → β} [T3Space β] {a : α} (hs : Dense s) (hf : ∀ᶠ (x : α) in nhds a, ∃ b, Filter.Tendsto f (Filter.comap Subtype.val (nhds x)) (nhds b)) : ContinuousAt (hs.extend f) a
Dense.extend_unique_at 📋 Mathlib.Topology.DenseEmbedding
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {s : Set α} {f : ↑s → β} [T2Space β] {a : α} {g : α → β} (hs : Dense s) (hf : ∀ᶠ (x : ↑s) in Filter.comap Subtype.val (nhds a), g ↑x = f x) (hg : ContinuousAt g a) : hs.extend f a = g a
continuous_of_mulTSupport 📋 Mathlib.Topology.Algebra.Support
{α : Type u_2} {β : Type u_4} [TopologicalSpace α] [One β] [TopologicalSpace β] {f : α → β} (hf : ∀ x ∈ mulTSupport f, ContinuousAt f x) : Continuous f
continuous_of_tsupport 📋 Mathlib.Topology.Algebra.Support
{α : Type u_2} {β : Type u_4} [TopologicalSpace α] [Zero β] [TopologicalSpace β] {f : α → β} (hf : ∀ x ∈ tsupport f, ContinuousAt f x) : Continuous f
ContinuousAt.fun_const_smul 📋 Mathlib.Topology.Algebra.ConstMulAction
{M : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [SMul M α] [ContinuousConstSMul M α] [TopologicalSpace β] {g : β → α} {b : β} (hg : ContinuousAt g b) (c : M) : ContinuousAt (fun i => c • g i) b
ContinuousAt.fun_const_vadd 📋 Mathlib.Topology.Algebra.ConstMulAction
{M : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [VAdd M α] [ContinuousConstVAdd M α] [TopologicalSpace β] {g : β → α} {b : β} (hg : ContinuousAt g b) (c : M) : ContinuousAt (fun i => c +ᵥ g i) b
ContinuousAt.const_smul 📋 Mathlib.Topology.Algebra.ConstMulAction
{M : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [SMul M α] [ContinuousConstSMul M α] [TopologicalSpace β] {g : β → α} {b : β} (hg : ContinuousAt g b) (c : M) : ContinuousAt (c • g) b
ContinuousAt.const_vadd 📋 Mathlib.Topology.Algebra.ConstMulAction
{M : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [VAdd M α] [ContinuousConstVAdd M α] [TopologicalSpace β] {g : β → α} {b : β} (hg : ContinuousAt g b) (c : M) : ContinuousAt (c +ᵥ g) b
IsUnit.continuousAt_const_smul_iff 📋 Mathlib.Topology.Algebra.ConstMulAction
{M : Type u_1} {α : Type u_2} {β : Type u_3} [Monoid M] [TopologicalSpace α] [MulAction M α] [ContinuousConstSMul M α] [TopologicalSpace β] {f : β → α} {b : β} {c : M} (hc : IsUnit c) : ContinuousAt (fun x => c • f x) b ↔ ContinuousAt f b
continuousAt_const_smul_iff 📋 Mathlib.Topology.Algebra.ConstMulAction
{α : Type u_2} {β : Type u_3} {G : Type u_4} [TopologicalSpace α] [Group G] [MulAction G α] [ContinuousConstSMul G α] [TopologicalSpace β] {f : β → α} {b : β} (c : G) : ContinuousAt (fun x => c • f x) b ↔ ContinuousAt f b
continuousAt_const_vadd_iff 📋 Mathlib.Topology.Algebra.ConstMulAction
{α : Type u_2} {β : Type u_3} {G : Type u_4} [TopologicalSpace α] [AddGroup G] [AddAction G α] [ContinuousConstVAdd G α] [TopologicalSpace β] {f : β → α} {b : β} (c : G) : ContinuousAt (fun x => c +ᵥ f x) b ↔ ContinuousAt f b
continuousAt_const_smul_iff₀ 📋 Mathlib.Topology.Algebra.ConstMulAction
{α : Type u_2} {β : Type u_3} {G₀ : Type u_4} [TopologicalSpace α] [GroupWithZero G₀] [MulAction G₀ α] [ContinuousConstSMul G₀ α] [TopologicalSpace β] {f : β → α} {b : β} {c : G₀} (hc : c ≠ 0) : ContinuousAt (fun x => c • f x) b ↔ ContinuousAt f b
ContinuousAt.add_const 📋 Mathlib.Topology.Algebra.Monoid.Defs
{M : Type u_1} [TopologicalSpace M] [Add M] [SeparatelyContinuousAdd M] {X : Type u_2} [TopologicalSpace X] {f : X → M} {x : X} (hf : ContinuousAt f x) (b : M) : ContinuousAt (fun x => f x + b) x
ContinuousAt.const_add 📋 Mathlib.Topology.Algebra.Monoid.Defs
{M : Type u_1} [TopologicalSpace M] [Add M] [SeparatelyContinuousAdd M] {X : Type u_2} [TopologicalSpace X] {f : X → M} {x : X} (hf : ContinuousAt f x) (b : M) : ContinuousAt (fun x => b + f x) x
ContinuousAt.const_mul 📋 Mathlib.Topology.Algebra.Monoid.Defs
{M : Type u_1} [TopologicalSpace M] [Mul M] [SeparatelyContinuousMul M] {X : Type u_2} [TopologicalSpace X] {f : X → M} {x : X} (hf : ContinuousAt f x) (b : M) : ContinuousAt (fun x => b * f x) x
ContinuousAt.mul_const 📋 Mathlib.Topology.Algebra.Monoid.Defs
{M : Type u_1} [TopologicalSpace M] [Mul M] [SeparatelyContinuousMul M] {X : Type u_2} [TopologicalSpace X] {f : X → M} {x : X} (hf : ContinuousAt f x) (b : M) : ContinuousAt (fun x => f x * b) x
ContinuousAt.fun_add 📋 Mathlib.Topology.Algebra.Monoid.Defs
{M : Type u_1} [TopologicalSpace M] [Add M] [ContinuousAdd M] {X : Type u_2} [TopologicalSpace X] {f g : X → M} {x : X} (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (fun i => f i + g i) x
ContinuousAt.fun_mul 📋 Mathlib.Topology.Algebra.Monoid.Defs
{M : Type u_1} [TopologicalSpace M] [Mul M] [ContinuousMul M] {X : Type u_2} [TopologicalSpace X] {f g : X → M} {x : X} (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (fun i => f i * g i) x
ContinuousAt.add 📋 Mathlib.Topology.Algebra.Monoid.Defs
{M : Type u_1} [TopologicalSpace M] [Add M] [ContinuousAdd M] {X : Type u_2} [TopologicalSpace X] {f g : X → M} {x : X} (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (f + g) x
ContinuousAt.mul 📋 Mathlib.Topology.Algebra.Monoid.Defs
{M : Type u_1} [TopologicalSpace M] [Mul M] [ContinuousMul M] {X : Type u_2} [TopologicalSpace X] {f g : X → M} {x : X} (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (f * g) x
ContinuousAt.fun_inv 📋 Mathlib.Topology.Algebra.Group.Defs
{G : Type u_1} {X : Type u_3} [TopologicalSpace X] [TopologicalSpace G] [Inv G] [ContinuousInv G] {f : X → G} {x : X} (hf : ContinuousAt f x) : ContinuousAt (fun i => (f i)⁻¹) x
ContinuousAt.fun_neg 📋 Mathlib.Topology.Algebra.Group.Defs
{G : Type u_1} {X : Type u_3} [TopologicalSpace X] [TopologicalSpace G] [Neg G] [ContinuousNeg G] {f : X → G} {x : X} (hf : ContinuousAt f x) : ContinuousAt (fun i => -f i) x
ContinuousAt.inv 📋 Mathlib.Topology.Algebra.Group.Defs
{G : Type u_1} {X : Type u_3} [TopologicalSpace X] [TopologicalSpace G] [Inv G] [ContinuousInv G] {f : X → G} {x : X} (hf : ContinuousAt f x) : ContinuousAt f⁻¹ x
ContinuousAt.neg 📋 Mathlib.Topology.Algebra.Group.Defs
{G : Type u_1} {X : Type u_3} [TopologicalSpace X] [TopologicalSpace G] [Neg G] [ContinuousNeg G] {f : X → G} {x : X} (hf : ContinuousAt f x) : ContinuousAt (-f) x
ContinuousAt.fun_div' 📋 Mathlib.Topology.Algebra.Group.Defs
{G : Type u_1} {X : Type u_3} [TopologicalSpace X] [TopologicalSpace G] [Div G] [ContinuousDiv G] {f g : X → G} {x : X} (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (fun i => f i / g i) x
ContinuousAt.fun_sub 📋 Mathlib.Topology.Algebra.Group.Defs
{G : Type u_1} {X : Type u_3} [TopologicalSpace X] [TopologicalSpace G] [Sub G] [ContinuousSub G] {f g : X → G} {x : X} (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (fun i => f i - g i) x
ContinuousAt.div' 📋 Mathlib.Topology.Algebra.Group.Defs
{G : Type u_1} {X : Type u_3} [TopologicalSpace X] [TopologicalSpace G] [Div G] [ContinuousDiv G] {f g : X → G} {x : X} (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (f / g) x
ContinuousAt.sub 📋 Mathlib.Topology.Algebra.Group.Defs
{G : Type u_1} {X : Type u_3} [TopologicalSpace X] [TopologicalSpace G] [Sub G] [ContinuousSub G] {f g : X → G} {x : X} (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (f - g) x
ContinuousAt.fun_smul 📋 Mathlib.Topology.Algebra.MulAction
{M : Type u_1} {X : Type u_2} {Y : Type u_3} [TopologicalSpace M] [TopologicalSpace X] [TopologicalSpace Y] [SMul M X] [ContinuousSMul M X] {f : Y → M} {g : Y → X} {b : Y} (hf : ContinuousAt f b) (hg : ContinuousAt g b) : ContinuousAt (fun i => f i • g i) b
ContinuousAt.fun_vadd 📋 Mathlib.Topology.Algebra.MulAction
{M : Type u_1} {X : Type u_2} {Y : Type u_3} [TopologicalSpace M] [TopologicalSpace X] [TopologicalSpace Y] [VAdd M X] [ContinuousVAdd M X] {f : Y → M} {g : Y → X} {b : Y} (hf : ContinuousAt f b) (hg : ContinuousAt g b) : ContinuousAt (fun i => f i +ᵥ g i) b
ContinuousAt.smul 📋 Mathlib.Topology.Algebra.MulAction
{M : Type u_1} {X : Type u_2} {Y : Type u_3} [TopologicalSpace M] [TopologicalSpace X] [TopologicalSpace Y] [SMul M X] [ContinuousSMul M X] {f : Y → M} {g : Y → X} {b : Y} (hf : ContinuousAt f b) (hg : ContinuousAt g b) : ContinuousAt (f • g) b
ContinuousAt.vadd 📋 Mathlib.Topology.Algebra.MulAction
{M : Type u_1} {X : Type u_2} {Y : Type u_3} [TopologicalSpace M] [TopologicalSpace X] [TopologicalSpace Y] [VAdd M X] [ContinuousVAdd M X] {f : Y → M} {g : Y → X} {b : Y} (hf : ContinuousAt f b) (hg : ContinuousAt g b) : ContinuousAt (f +ᵥ g) b
map_continuousAt 📋 Mathlib.Topology.ContinuousMap.Basic
{F : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] [ContinuousMapClass F α β] (f : F) (a : α) : ContinuousAt (⇑f) a
ContinuousMap.continuousAt 📋 Mathlib.Topology.ContinuousMap.Basic
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) (x : α) : ContinuousAt (⇑f) x
continuousAt_nsmul 📋 Mathlib.Topology.Algebra.Monoid
{M : Type u_3} [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] (x : M) (n : ℕ) : ContinuousAt (fun x => n • x) x
continuousAt_pow 📋 Mathlib.Topology.Algebra.Monoid
{M : Type u_3} [TopologicalSpace M] [Monoid M] [ContinuousMul M] (x : M) (n : ℕ) : ContinuousAt (fun x => x ^ n) x
ContinuousAt.fun_nsmul 📋 Mathlib.Topology.Algebra.Monoid
{M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {f : X → M} {x : X} (hf : ContinuousAt f x) (n : ℕ) : ContinuousAt (fun i => n • f i) x
ContinuousAt.fun_pow 📋 Mathlib.Topology.Algebra.Monoid
{M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [Monoid M] [ContinuousMul M] {f : X → M} {x : X} (hf : ContinuousAt f x) (n : ℕ) : ContinuousAt (fun i => f i ^ n) x
ContinuousAt.nsmul 📋 Mathlib.Topology.Algebra.Monoid
{M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddMonoid M] [ContinuousAdd M] {f : X → M} {x : X} (hf : ContinuousAt f x) (n : ℕ) : ContinuousAt (n • f) x
ContinuousAt.pow 📋 Mathlib.Topology.Algebra.Monoid
{M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [Monoid M] [ContinuousMul M] {f : X → M} {x : X} (hf : ContinuousAt f x) (n : ℕ) : ContinuousAt (f ^ n) x
continuousAt_zpow 📋 Mathlib.Topology.Algebra.Group.ZPow
{G : Type u_1} [TopologicalSpace G] [Group G] [IsTopologicalGroup G] (x : G) (z : ℤ) : ContinuousAt (fun x => x ^ z) x
continuousAt_zsmul 📋 Mathlib.Topology.Algebra.Group.ZPow
{G : Type u_1} [TopologicalSpace G] [AddGroup G] [IsTopologicalAddGroup G] (x : G) (z : ℤ) : ContinuousAt (fun x => z • x) x
ContinuousAt.fun_zpow 📋 Mathlib.Topology.Algebra.Group.ZPow
{G : Type u_1} {α : Type u_2} [TopologicalSpace G] [Group G] [IsTopologicalGroup G] [TopologicalSpace α] {f : α → G} {x : α} (hf : ContinuousAt f x) (z : ℤ) : ContinuousAt (fun i => f i ^ z) x
ContinuousAt.fun_zsmul 📋 Mathlib.Topology.Algebra.Group.ZPow
{G : Type u_1} {α : Type u_2} [TopologicalSpace G] [AddGroup G] [IsTopologicalAddGroup G] [TopologicalSpace α] {f : α → G} {x : α} (hf : ContinuousAt f x) (z : ℤ) : ContinuousAt (fun i => z • f i) x
ContinuousAt.zpow 📋 Mathlib.Topology.Algebra.Group.ZPow
{G : Type u_1} {α : Type u_2} [TopologicalSpace G] [Group G] [IsTopologicalGroup G] [TopologicalSpace α] {f : α → G} {x : α} (hf : ContinuousAt f x) (z : ℤ) : ContinuousAt (f ^ z) x
ContinuousAt.zsmul 📋 Mathlib.Topology.Algebra.Group.ZPow
{G : Type u_1} {α : Type u_2} [TopologicalSpace G] [AddGroup G] [IsTopologicalAddGroup G] [TopologicalSpace α] {f : α → G} {x : α} (hf : ContinuousAt f x) (z : ℤ) : ContinuousAt (z • f) x
Uniform.continuousAt_iff'_left 📋 Mathlib.Topology.UniformSpace.Basic
{α : Type ua} {β : Type ub} [UniformSpace α] [TopologicalSpace β] {f : β → α} {b : β} : ContinuousAt f b ↔ Filter.Tendsto (fun x => (f x, f b)) (nhds b) (uniformity α)
Uniform.continuousAt_iff'_right 📋 Mathlib.Topology.UniformSpace.Basic
{α : Type ua} {β : Type ub} [UniformSpace α] [TopologicalSpace β] {f : β → α} {b : β} : ContinuousAt f b ↔ Filter.Tendsto (fun x => (f b, f x)) (nhds b) (uniformity α)
Uniform.continuousAt_iff_prod 📋 Mathlib.Topology.UniformSpace.Basic
{α : Type ua} {β : Type ub} [UniformSpace α] [TopologicalSpace β] {f : β → α} {b : β} : ContinuousAt f b ↔ Filter.Tendsto (fun x => (f x.1, f x.2)) (nhds (b, b)) (uniformity α)
Uniform.exists_is_open_mem_uniformity_of_forall_mem_eq 📋 Mathlib.Topology.UniformSpace.Basic
{α : Type ua} {β : Type ub} [UniformSpace α] [TopologicalSpace β] {r : SetRel α α} {s : Set β} {f g : β → α} (hf : ∀ x ∈ s, ContinuousAt f x) (hg : ∀ x ∈ s, ContinuousAt g x) (hfg : Set.EqOn f g s) (hr : r ∈ uniformity α) : ∃ t, IsOpen t ∧ s ⊆ t ∧ ∀ x ∈ t, (f x, g x) ∈ r
TendstoUniformly.tendsto_comp 📋 Mathlib.Topology.UniformSpace.UniformApproximation
{α : Type u_1} {β : Type u_2} {ι : Type u_3} [TopologicalSpace α] [UniformSpace β] {F : ι → α → β} {f : α → β} {x : α} {p : Filter ι} {g : ι → α} (h : TendstoUniformly F f p) (hf : ContinuousAt f x) (hg : Filter.Tendsto g p (nhds x)) : Filter.Tendsto (fun n => F n (g n)) p (nhds (f x))
TendstoLocallyUniformly.tendsto_comp 📋 Mathlib.Topology.UniformSpace.UniformApproximation
{α : Type u_1} {β : Type u_2} {ι : Type u_3} [TopologicalSpace α] [UniformSpace β] {F : ι → α → β} {f : α → β} {x : α} {p : Filter ι} {g : ι → α} (h : TendstoLocallyUniformly F f p) (hf : ContinuousAt f x) (hg : Filter.Tendsto g p (nhds x)) : Filter.Tendsto (fun n => F n (g n)) p (nhds (f x))
continuous_of_locally_uniform_approx_of_continuousAt 📋 Mathlib.Topology.UniformSpace.UniformApproximation
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [UniformSpace β] {f : α → β} (L : ∀ (x : α), ∀ u ∈ uniformity β, ∃ t ∈ nhds x, ∃ F, ContinuousAt F x ∧ ∀ y ∈ t, (f y, F y) ∈ u) : Continuous f
continuousAt_of_locally_uniform_approx_of_continuousAt 📋 Mathlib.Topology.UniformSpace.UniformApproximation
{α : Type u_1} {β : Type u_2} [TopologicalSpace α] [UniformSpace β] {f : α → β} {x : α} (L : ∀ u ∈ uniformity β, ∃ t ∈ nhds x, ∃ F, ContinuousAt F x ∧ ∀ y ∈ t, (f y, F y) ∈ u) : ContinuousAt f x
tendsto_comp_of_locally_uniform_limit 📋 Mathlib.Topology.UniformSpace.UniformApproximation
{α : Type u_1} {β : Type u_2} {ι : Type u_3} [TopologicalSpace α] [UniformSpace β] {F : ι → α → β} {f : α → β} {x : α} {p : Filter ι} {g : ι → α} (h : ContinuousAt f x) (hg : Filter.Tendsto g p (nhds x)) (hunif : ∀ u ∈ uniformity β, ∃ t ∈ nhds x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u) : Filter.Tendsto (fun n => F n (g n)) p (nhds (f x))
UniformOnFun.continuousAt_eval₂ 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
{α : Type u_1} {β : Type u_2} [UniformSpace β] {𝔖 : Set (Set α)} [TopologicalSpace α] {f : UniformOnFun α β 𝔖} {x : α} (h𝔖 : ∃ V ∈ 𝔖, V ∈ nhds x) (hc : ContinuousAt ((UniformOnFun.toFun 𝔖) f) x) : ContinuousAt (fun fx => (UniformOnFun.toFun 𝔖) fx.1 fx.2) (f, x)
UniformOnFun.continuousOn_eval₂ 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
{α : Type u_1} {β : Type u_2} [UniformSpace β] {𝔖 : Set (Set α)} [TopologicalSpace α] (h𝔖 : ∀ (x : α), ∃ V ∈ 𝔖, V ∈ nhds x) : ContinuousOn (fun fx => (UniformOnFun.toFun 𝔖) fx.1 fx.2) {fx | ContinuousAt ((UniformOnFun.toFun 𝔖) fx.1) fx.2}
EquicontinuousAt.continuousAt 📋 Mathlib.Topology.UniformSpace.Equicontinuity
{ι : Type u_1} {X : Type u_3} {α : Type u_6} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α} {x₀ : X} (h : EquicontinuousAt F x₀) (i : ι) : ContinuousAt (F i) x₀
equicontinuousAt_finite 📋 Mathlib.Topology.UniformSpace.Equicontinuity
{ι : Type u_1} {X : Type u_3} {α : Type u_6} [tX : TopologicalSpace X] [uα : UniformSpace α] [Finite ι] {F : ι → X → α} {x₀ : X} : EquicontinuousAt F x₀ ↔ ∀ (i : ι), ContinuousAt (F i) x₀
equicontinuousAt_unique 📋 Mathlib.Topology.UniformSpace.Equicontinuity
{ι : Type u_1} {X : Type u_3} {α : Type u_6} [tX : TopologicalSpace X] [uα : UniformSpace α] [Unique ι] {F : ι → X → α} {x : X} : EquicontinuousAt F x ↔ ContinuousAt (F default) x
Set.EquicontinuousAt.continuousAt_of_mem 📋 Mathlib.Topology.UniformSpace.Equicontinuity
{X : Type u_3} {α : Type u_6} [tX : TopologicalSpace X] [uα : UniformSpace α] {H : Set (X → α)} {x₀ : X} (h : H.EquicontinuousAt x₀) {f : X → α} (hf : f ∈ H) : ContinuousAt f x₀
Filter.Tendsto.continuousAt_of_equicontinuousAt 📋 Mathlib.Topology.UniformSpace.Equicontinuity
{ι : Type u_1} {X : Type u_3} {α : Type u_6} [tX : TopologicalSpace X] [uα : UniformSpace α] {l : Filter ι} [l.NeBot] {F : ι → X → α} {f : X → α} {x₀ : X} (h₁ : Filter.Tendsto F l (nhds f)) (h₂ : EquicontinuousAt F x₀) : ContinuousAt f x₀
equicontinuousAt_iff_continuousAt 📋 Mathlib.Topology.UniformSpace.Equicontinuity
{ι : Type u_1} {X : Type u_3} {α : Type u_6} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α} {x₀ : X} : EquicontinuousAt F x₀ ↔ ContinuousAt (⇑UniformFun.ofFun ∘ Function.swap F) x₀
IsCompact.uniformContinuousAt_of_continuousAt 📋 Mathlib.Topology.UniformSpace.HeineCantor
{α : Type u_1} {β : Type u_2} [UniformSpace α] [UniformSpace β] {r : Set (β × β)} {s : Set α} (hs : IsCompact s) (f : α → β) (hf : ∀ a ∈ s, ContinuousAt f a) (hr : r ∈ uniformity β) : {x | x.1 ∈ s → (f x.1, f x.2) ∈ r} ∈ uniformity α
continuousAt_inv 📋 Mathlib.Topology.Algebra.Group.ContinuousInv
{G : Type u_1} [TopologicalSpace G] [Inv G] [ContinuousInv G] {x : G} : ContinuousAt Inv.inv x
continuousAt_neg 📋 Mathlib.Topology.Algebra.Group.ContinuousInv
{G : Type u_1} [TopologicalSpace G] [Neg G] [ContinuousNeg G] {x : G} : ContinuousAt Neg.neg x
ContinuousAt.of_inv 📋 Mathlib.Topology.Algebra.Group.ContinuousInv
{G : Type u_1} {α : Type u_3} [TopologicalSpace G] [InvolutiveInv G] [ContinuousInv G] [TopologicalSpace α] {f : α → G} {x : α} : ContinuousAt f⁻¹ x → ContinuousAt f x
ContinuousAt.of_neg 📋 Mathlib.Topology.Algebra.Group.ContinuousInv
{G : Type u_1} {α : Type u_3} [TopologicalSpace G] [InvolutiveNeg G] [ContinuousNeg G] [TopologicalSpace α] {f : α → G} {x : α} : ContinuousAt (-f) x → ContinuousAt f x
continuousAt_inv_iff 📋 Mathlib.Topology.Algebra.Group.ContinuousInv
{G : Type u_1} {α : Type u_3} [TopologicalSpace G] [InvolutiveInv G] [ContinuousInv G] [TopologicalSpace α] {f : α → G} {x : α} : ContinuousAt f⁻¹ x ↔ ContinuousAt f x
continuousAt_neg_iff 📋 Mathlib.Topology.Algebra.Group.ContinuousInv
{G : Type u_1} {α : Type u_3} [TopologicalSpace G] [InvolutiveNeg G] [ContinuousNeg G] [TopologicalSpace α] {f : α → G} {x : α} : ContinuousAt (-f) x ↔ ContinuousAt f x
continuous_of_continuousAt_one 📋 Mathlib.Topology.Algebra.Group.Neighborhood
{G : Type u_1} [TopologicalSpace G] [Group G] [IsTopologicalGroup G] {M : Type u_2} {hom : Type u_3} [MulOneClass M] [TopologicalSpace M] [ContinuousMul M] [FunLike hom G M] [MonoidHomClass hom G M] (f : hom) (hf : ContinuousAt (⇑f) 1) : Continuous ⇑f
continuous_of_continuousAt_zero 📋 Mathlib.Topology.Algebra.Group.Neighborhood
{G : Type u_1} [TopologicalSpace G] [AddGroup G] [IsTopologicalAddGroup G] {M : Type u_2} {hom : Type u_3} [AddZeroClass M] [TopologicalSpace M] [ContinuousAdd M] [FunLike hom G M] [AddMonoidHomClass hom G M] (f : hom) (hf : ContinuousAt (⇑f) 0) : Continuous ⇑f
continuous_of_continuousAt_one₂ 📋 Mathlib.Topology.Algebra.Group.Neighborhood
{G : Type u_1} [TopologicalSpace G] [Group G] [IsTopologicalGroup G] {H : Type u_2} {M : Type u_3} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M] [Group H] [TopologicalSpace H] [IsTopologicalGroup H] (f : G →* H →* M) (hf : ContinuousAt (fun x => (f x.1) x.2) (1, 1)) (hl : ∀ (x : G), ContinuousAt (⇑(f x)) 1) (hr : ∀ (y : H), ContinuousAt (fun x => (f x) y) 1) : Continuous fun x => (f x.1) x.2
continuous_of_continuousAt_zero₂ 📋 Mathlib.Topology.Algebra.Group.Neighborhood
{G : Type u_1} [TopologicalSpace G] [AddGroup G] [IsTopologicalAddGroup G] {H : Type u_2} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] [ContinuousAdd M] [AddGroup H] [TopologicalSpace H] [IsTopologicalAddGroup H] (f : G →+ H →+ M) (hf : ContinuousAt (fun x => (f x.1) x.2) (0, 0)) (hl : ∀ (x : G), ContinuousAt (⇑(f x)) 0) (hr : ∀ (y : H), ContinuousAt (fun x => (f x) y) 0) : Continuous fun x => (f x.1) x.2
uniformContinuous_of_continuousAt_one 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
{α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [IsUniformGroup α] {hom : Type u_3} [UniformSpace β] [Group β] [IsUniformGroup β] [FunLike hom α β] [MonoidHomClass hom α β] (f : hom) (hf : ContinuousAt (⇑f) 1) : UniformContinuous ⇑f
uniformContinuous_of_continuousAt_zero 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
{α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] {hom : Type u_3} [UniformSpace β] [AddGroup β] [IsUniformAddGroup β] [FunLike hom α β] [AddMonoidHomClass hom α β] (f : hom) (hf : ContinuousAt (⇑f) 0) : UniformContinuous ⇑f
AddMonoidHom.uniformContinuous_of_continuousAt_zero 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
{α : Type u_1} {β : Type u_2} [UniformSpace α] [AddGroup α] [IsUniformAddGroup α] [UniformSpace β] [AddGroup β] [IsUniformAddGroup β] (f : α →+ β) (hf : ContinuousAt (⇑f) 0) : UniformContinuous ⇑f
MonoidHom.uniformContinuous_of_continuousAt_one 📋 Mathlib.Topology.Algebra.IsUniformGroup.Defs
{α : Type u_1} {β : Type u_2} [UniformSpace α] [Group α] [IsUniformGroup α] [UniformSpace β] [Group β] [IsUniformGroup β] (f : α →* β) (hf : ContinuousAt (⇑f) 1) : UniformContinuous ⇑f
ContinuousInv₀.continuousAt_inv₀ 📋 Mathlib.Topology.Algebra.GroupWithZero
{G₀ : Type u_4} {inst✝ : Zero G₀} {inst✝¹ : Inv G₀} {inst✝² : TopologicalSpace G₀} [self : ContinuousInv₀ G₀] ⦃x : G₀⦄ : x ≠ 0 → ContinuousAt Inv.inv x
ContinuousInv₀.mk 📋 Mathlib.Topology.Algebra.GroupWithZero
{G₀ : Type u_4} [Zero G₀] [Inv G₀] [TopologicalSpace G₀] (continuousAt_inv₀ : ∀ ⦃x : G₀⦄, x ≠ 0 → ContinuousAt Inv.inv x) : ContinuousInv₀ G₀
ContinuousAt.fun_inv₀ 📋 Mathlib.Topology.Algebra.GroupWithZero
{α : Type u_1} {G₀ : Type u_3} [Zero G₀] [Inv G₀] [TopologicalSpace G₀] [ContinuousInv₀ G₀] {f : α → G₀} {a : α} [TopologicalSpace α] (hf : ContinuousAt f a) (ha : f a ≠ 0) : ContinuousAt (fun i => (f i)⁻¹) a
ContinuousAt.div_const 📋 Mathlib.Topology.Algebra.GroupWithZero
{α : Type u_1} {G₀ : Type u_3} [DivInvMonoid G₀] [TopologicalSpace G₀] [SeparatelyContinuousMul G₀] {f : α → G₀} [TopologicalSpace α] {a : α} (hf : ContinuousAt f a) (y : G₀) : ContinuousAt (fun x => f x / y) a
ContinuousAt.inv₀ 📋 Mathlib.Topology.Algebra.GroupWithZero
{α : Type u_1} {G₀ : Type u_3} [Zero G₀] [Inv G₀] [TopologicalSpace G₀] [ContinuousInv₀ G₀] {f : α → G₀} {a : α} [TopologicalSpace α] (hf : ContinuousAt f a) (ha : f a ≠ 0) : ContinuousAt f⁻¹ a
continuousAt_zpow₀ 📋 Mathlib.Topology.Algebra.GroupWithZero
{G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [ContinuousInv₀ G₀] [ContinuousMul G₀] (x : G₀) (m : ℤ) (h : x ≠ 0 ∨ 0 ≤ m) : ContinuousAt (fun x => x ^ m) x
ContinuousAt.div₀ 📋 Mathlib.Topology.Algebra.GroupWithZero
{α : Type u_1} {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [ContinuousInv₀ G₀] [ContinuousMul G₀] {f g : α → G₀} [TopologicalSpace α] {a : α} (hf : ContinuousAt f a) (hg : ContinuousAt g a) (h₀ : g a ≠ 0) : ContinuousAt (fun x => f x / g x) a
ContinuousAt.zpow₀ 📋 Mathlib.Topology.Algebra.GroupWithZero
{G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [ContinuousInv₀ G₀] [ContinuousMul G₀] {X : Type u_4} [TopologicalSpace X] {a : X} {f : X → G₀} (hf : ContinuousAt f a) (m : ℤ) (h : f a ≠ 0 ∨ 0 ≤ m) : ContinuousAt (fun x => f x ^ m) a
ContinuousAt.div 📋 Mathlib.Topology.Algebra.GroupWithZero
{α : Type u_1} {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [ContinuousInv₀ G₀] [ContinuousMul G₀] {f g : α → G₀} [TopologicalSpace α] {a : α} (hf : ContinuousAt f a) (hg : ContinuousAt g a) (h₀ : g a ≠ 0) : ContinuousAt (f / g) a
Continuous.comp_div_cases 📋 Mathlib.Topology.Algebra.GroupWithZero
{α : Type u_1} {β : Type u_2} {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [ContinuousInv₀ G₀] [ContinuousMul G₀] [TopologicalSpace α] [TopologicalSpace β] {f g : α → G₀} (h : α → G₀ → β) (hf : Continuous f) (hg : Continuous g) (hh : ∀ (a : α), g a ≠ 0 → ContinuousAt ↿h (a, f a / g a)) (h2h : ∀ (a : α), g a = 0 → Filter.Tendsto (↿h) (nhds a ×ˢ ⊤) (nhds (h a 0))) : Continuous fun x => h x (f x / g x)
ContinuousAt.comp_div_cases 📋 Mathlib.Topology.Algebra.GroupWithZero
{α : Type u_1} {β : Type u_2} {G₀ : Type u_3} [GroupWithZero G₀] [TopologicalSpace G₀] [ContinuousInv₀ G₀] [ContinuousMul G₀] [TopologicalSpace α] [TopologicalSpace β] {a : α} {f g : α → G₀} (h : α → G₀ → β) (hf : ContinuousAt f a) (hg : ContinuousAt g a) (hh : g a ≠ 0 → ContinuousAt ↿h (a, f a / g a)) (h2h : g a = 0 → Filter.Tendsto (↿h) (nhds a ×ˢ ⊤) (nhds (h a 0))) : ContinuousAt (fun x => h x (f x / g x)) a
IsLocalExtr.comp_continuous 📋 Mathlib.Topology.Order.LocalExtr
{α : Type u} {β : Type v} {δ : Type x} [TopologicalSpace α] [Preorder β] {f : α → β} [TopologicalSpace δ] {g : δ → α} {b : δ} (hf : IsLocalExtr f (g b)) (hg : ContinuousAt g b) : IsLocalExtr (f ∘ g) b
IsLocalMax.comp_continuous 📋 Mathlib.Topology.Order.LocalExtr
{α : Type u} {β : Type v} {δ : Type x} [TopologicalSpace α] [Preorder β] {f : α → β} [TopologicalSpace δ] {g : δ → α} {b : δ} (hf : IsLocalMax f (g b)) (hg : ContinuousAt g b) : IsLocalMax (f ∘ g) b
IsLocalMin.comp_continuous 📋 Mathlib.Topology.Order.LocalExtr
{α : Type u} {β : Type v} {δ : Type x} [TopologicalSpace α] [Preorder β] {f : α → β} [TopologicalSpace δ] {g : δ → α} {b : δ} (hf : IsLocalMin f (g b)) (hg : ContinuousAt g b) : IsLocalMin (f ∘ g) b
ContinuousAt.abs 📋 Mathlib.Topology.Algebra.Order.Group
{G : Type u_1} [TopologicalSpace G] [AddCommGroup G] [LinearOrder G] [IsOrderedAddMonoid G] [OrderTopology G] {X : Type u_2} [TopologicalSpace X] {f : X → G} {x : X} (h : ContinuousAt f x) : ContinuousAt (fun x => |f x|) x
ContinuousAt.mabs 📋 Mathlib.Topology.Algebra.Order.Group
{G : Type u_1} [TopologicalSpace G] [CommGroup G] [LinearOrder G] [IsOrderedMonoid G] [OrderTopology G] {X : Type u_2} [TopologicalSpace X] {f : X → G} {x : X} (h : ContinuousAt f x) : ContinuousAt (fun x => |f x|ₘ) x
continuousAt_toIcoDiv 📋 Mathlib.Topology.Instances.AddCircle.Defs
{𝕜 : Type u_1} [AddCommGroup 𝕜] [LinearOrder 𝕜] [IsOrderedAddMonoid 𝕜] [Archimedean 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p : 𝕜} (hp : 0 < p) (a : 𝕜) {x : 𝕜} (hx : ¬x ≡ a [PMOD p]) : ContinuousAt (toIcoDiv hp a) x
continuousAt_toIcoMod 📋 Mathlib.Topology.Instances.AddCircle.Defs
{𝕜 : Type u_1} [AddCommGroup 𝕜] [LinearOrder 𝕜] [IsOrderedAddMonoid 𝕜] [Archimedean 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p : 𝕜} (hp : 0 < p) (a : 𝕜) {x : 𝕜} (hx : ¬x ≡ a [PMOD p]) : ContinuousAt (toIcoMod hp a) x

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

By constant:
🔍 Real.sin
finds all lemmas whose statement somehow mentions the sine function.
By lemma name substring:
🔍 "differ"
finds all lemmas that have "differ" somewhere in their lemma name.
By subexpression:
🔍 _ * (_ ^ _)
finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

The pattern can also be non-linear, as in
🔍 Real.sqrt ?a * Real.sqrt ?a

If the pattern has parameters, they are matched in any order. Both of these will find List.map:
🔍 (?a -> ?b) -> List ?a -> List ?b
🔍 List ?a -> (?a -> ?b) -> List ?b
By main conclusion:
🔍 |- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
🔍 |- _ < _ → tsum _ < tsum _
will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.
You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision 69fae59